Let Q be an orthogonal matrix. (a) Prove that if λis an eigenvalue, then so is 1 / λ. (b) Prove

step 1 So we have Q as an orthogonal matrix. By definition of orthogonal matrix, Q transpose into Q is

Let Q be an orthogonal matrix. (a) Prove that if λis an...

Let $Q$ be an orthogonal matrix. (a) Prove that if $\lambda$ is an eigenvalue, then so is $1 / \lambda$. (b) Prove that all its eigenvalues are complex numbers of modulus $|\lambda|=1$. In particular, the only possible real eigenvalues of an orthogonal matrix are \pm 1 . (c) Suppose $\mathbf{v}=\mathbf{x}+i \mathbf{y}$ is a complex eigenvector corresponding to a non-real eigenvalue. Prove that its real and imaginary parts are orthogonal vectors having the same Euclidean norm.

Let $Q$ be an orthogonal matrix. (a) Prove that if $\lambda$ is an eigenvalue, then so is $1 / \lambda$.
(b) Prove that all its eigenvalues are complex numbers of modulus $|\lambda|=1$. In particular, the only possible real eigenvalues of an orthogonal matrix are \pm 1 . (c) Suppose $\mathbf{v}=\mathbf{x}+i \mathbf{y}$ is a complex eigenvector corresponding to a non-real eigenvalue. Prove that its real and imaginary parts are orthogonal vectors having the same Euclidean norm.

Key Concepts

Orthogonal Matrices

Orthogonal matrices are square matrices whose columns (and rows) form an orthonormal set. This means that an orthogonal matrix Q satisfies Q^T Q = I, where Q^T is the transpose of Q and I is the identity matrix. This characteristic ensures that the transformation Q preserves Euclidean lengths and angles, making them fundamental in understanding rotations and reflections in space.

Eigenvalue Properties

The eigenvalues of a matrix are the scalars ? such that there exists a nonzero vector v (called an eigenvector) satisfying Qv = ?v. For orthogonal matrices, an important property is that if ? is an eigenvalue, so is 1/?. This reciprocal pairing is a consequence of the invertibility of orthogonal matrices (with Q^{-1} = Q^T) and stems from fundamental matrix algebra identities applied to eigenvalue equations.

Unit Modulus of Eigenvalues

Because orthogonal matrices preserve lengths, if Qv = ?v for some nonzero vector v, then the norm of v and the norm of Qv must be identical. This forces the modulus of any eigenvalue ? to be 1, i.e., |?| = 1, meaning that eigenvalues of orthogonal matrices lie on the unit circle in the complex plane. This further restricts real eigenvalues to ±1.

Complex Eigenvectors and Their Real and Imaginary Parts

When an eigenvalue of an orthogonal matrix is non-real, its corresponding eigenvector is complex and can be expressed as v = x + iy with real vectors x and y. Analysis shows that these real and imaginary components have equal lengths and are orthogonal to each other. This separation into orthogonal components is a useful property that helps in understanding the geometric implications of complex eigenvalues in real vector spaces.

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